Stochastic oscillations in models of epidemics on a network of cities

Centro de Física da Matéria Condensada and Departamento de Física, Faculdade de Ciências da Universidade de Lisboa, Lisboa Codex, Portugal.
Physical Review E (Impact Factor: 2.29). 11/2011; 84(5 Pt 1):051919. DOI: 10.1103/PhysRevE.84.051919
Source: PubMed


We carry out an analytic investigation of stochastic oscillations in a susceptible-infected-recovered model of disease spread on a network of n cities. In the model a fraction f(jk) of individuals from city k commute to city j, where they may infect, or be infected by, others. Starting from a continuous-time Markov description of the model the deterministic equations, which are valid in the limit when the population of each city is infinite, are recovered. The stochastic fluctuations about the fixed point of these equations are derived by use of the van Kampen system-size expansion. The fixed point structure of the deterministic equations is remarkably simple: A unique nontrivial fixed point always exists and has the feature that the fraction of susceptible, infected, and recovered individuals is the same for each city irrespective of its size. We find that the stochastic fluctuations have an analogously simple dynamics: All oscillations have a single frequency, equal to that found in the one-city case. We interpret this phenomenon in terms of the properties of the spectrum of the matrix of the linear approximation of the deterministic equations at the fixed point.

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    • "We use the method known in the physics literature [14] as the van Kampen or system size (total population size, N ) expansion. The discussion of this standard method in the context of compartmental epidemiological models such as the SIR model we consider here or other related models of infectious diseases can be found in [1] [5] [10] [12] [11]. The most recent description of the method in application to the SIRSI model is given in the supplemental information to Ref. [11]. "

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    ABSTRACT: We study the synchronisation and phase-lag of fluctuations in the number of infected individuals in a network of cities between which individuals commute. The frequency and amplitude of these oscillations is known to be very well captured by the van Kampen system-size expansion, and we use this approximation to compute the complex coherence function that describes their correlation. We find that, if the infection rate differs from city to city and the coupling between them is not too strong, these oscillations are synchronised with a well defined phase lag between cities. The analytic description of the effect is shown to be in good agreement with the results of stochastic simulations for realistic population sizes.
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